id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-2f19aa5d7245d1d3f9b1bfcb5a7d49b06a5e312912b59dc82bdc2ca55a596469 | Alice and Bob play a game where they alternately write a number. A player loses if they cannot write a number that is not a linear combination of the numbers already written, using nonnegative integers. Alice plays first, and they cannot write the number $1$. Which player has a winning strategy? | [
"Alice has a winning strategy."
] | [
"Alice and Bob play a game where they alternately write a number. A player loses if they cannot write a number that is not a linear combination of the numbers already written, using nonnegative integers. Alice plays first, and they cannot write the number $1$. Which player has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 7
} | 14 | 0 | 0 |
omr-ws-9a22d402753c25d263c5409e03d24eb60a951c2895c361407c3e91b4bd8510d7 | Alice and Bob play a game where they first collectively agree on a three-letter word. Then, Alice writes on individual pieces of paper every single possible way to rearrange the letters in that word (excluding the original arrangement of letters). All of these pieces of paper are put into a hat and then one of these pi... | [
"\\(\\frac{109}{169}\\)"
] | [
"Alice and Bob play a game where they first collectively agree on a three-letter word. Then, Alice writes on individual pieces of paper every single possible way to rearrange the letters in that word (excluding the original arrangement of letters). All of these pieces of paper are put into a hat and then one of the... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06be8c23a45d17072497a4c74490c1587bf7e93e257dea71738d88f7169df259 | Alice and Bob play a game where they start with nonnegative integers \( m \) and \( n \) on a blackboard. On each turn, a player erases one of the integers and replaces it with a lesser nonnegative integer that has not been written before. Alice goes first. For which pairs \((m, n)\) does Alice have a winning strategy? | [
"Alice has a winning strategy for all pairs \\((m, n)\\) except \\((2k, 2k)\\) and \\((2k+1, 2k)\\) and their rearrangements, where \\(k \\ge 0\\)."
] | [
"Alice and Bob play a game where they start with nonnegative integers \\( m \\) and \\( n \\) on a blackboard. On each turn, a player erases one of the integers and replaces it with a lesser nonnegative integer that has not been written before. Alice goes first. For which pairs \\((m, n)\\) does Alice have a winnin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 5,
"tir": 7
} | 34 | 0 | 0 |
omr-ws-753e92b9c76d7580f13776c7d0c4debd1319e8cef9012298b4e7029826fd29ed | Alice and Bob play a game where they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shar... | [
"4"
] | [
"Alice and Bob play a game where they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ccdb7d3efbdd17d1705689b5d3b5aa77da60efc853c586a127e1d70d517506df | Alice and Bob play a game where they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment shar... | [
"6"
] | [
"Alice and Bob play a game where they take turns drawing segments of length $1$ in the Euclidean plane. Alice begins, drawing the first segment, and from then on, each segment must start at the endpoint of the previous segment. It is not permitted to draw the segment lying over the preceding one. If the new segment... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-00f7f61b58d1de1e028d32994b1ddde0e8102907ab5d04802558c2cd94e024cd | Alice and Bob play a game where they take turns removing stones from a heap that initially has \( n \) stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Define \( g(n) \) as the smallest positive integer \( k \) ... | [
"79"
] | [
"Alice and Bob play a game where they take turns removing stones from a heap that initially has \\( n \\) stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Define \\( g(n) \\) as the smallest positive integer... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-e6ff2bc43d00e6b5c189e9ad05b48b587831f2030c85279a74dc06ddf7eae2dd | Alice and Bob play a game where they take turns removing stones from a heap that initially has \( n \) stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Let \( W(k) \) be the smallest \( n \) such that Bob has a ... | [
"no_valid_answer_found"
] | [
"Alice and Bob play a game where they take turns removing stones from a heap that initially has \\( n \\) stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Let \\( W(k) \\) be the smallest \\( n \\) such that... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90751bcbc8efcaa38d9a464d1c46136fdf4ad96b457bdef33213f5742f8b8ceb | Alice and Bob play a game where they write fractions of the form \( \frac{1}{n} \) on a blackboard, with \( n \) being a positive integer. Alice writes one fraction per turn, while Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn, and so on. Bob's goal is to... | [
"2"
] | [
"Alice and Bob play a game where they write fractions of the form \\( \\frac{1}{n} \\) on a blackboard, with \\( n \\) being a positive integer. Alice writes one fraction per turn, while Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn, and so on. Bob's ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5492f2355a32aea80e3733519a467ada37ffea2183f8af620e32397a2ba6f8b1 | Alice and Bob play a game with 100 gold coins, 100 silver coins, and 100 bronze coins. Players take turns to take at least one coin, but they cannot take two or more coins of the same kind at once. Alice goes first. The player who cannot take any coin loses. Who has a winning strategy? | [
"Bob has a winning strategy."
] | [
"Alice and Bob play a game with 100 gold coins, 100 silver coins, and 100 bronze coins. Players take turns to take at least one coin, but they cannot take two or more coins of the same kind at once. Alice goes first. The player who cannot take any coin loses. Who has a winning strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 17
} | 39 | 0 | 0 |
omr-ws-1b37b6cd25cc18eb49fd60ea11dfc6170e6fe7b04d4a621ff729b7999f65cfe1 | Alice and Bob play a game with 101 hats arranged in a circle. Bob chooses a positive integer \( n \) and places a rabbit under one of the hats. Alice can choose a hat to check each time. If the rabbit is not under the chosen hat, Bob moves the rabbit to the \( n \)th hat in a clockwise direction. Let \( S \) be the set... | [
"5151"
] | [
"Alice and Bob play a game with 101 hats arranged in a circle. Bob chooses a positive integer \\( n \\) and places a rabbit under one of the hats. Alice can choose a hat to check each time. If the rabbit is not under the chosen hat, Bob moves the rabbit to the \\( n \\)th hat in a clockwise direction. Let \\( S \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-1e5ab88f1910d05a77513e162499003b73dc57219b95968727344414efed36bc | Alice and Bob play a game with a $3 \times 3$ grid containing the positive integers $1, 2, \dots, 9$ arranged randomly. On a player's turn, they select one of the four $2 \times 2$ squares of cells $S$ in the grid, containing at least two distinct numbers, and each number in $S$ is replaced by the smallest number in $S... | [
"47"
] | [
"Alice and Bob play a game with a $3 \\times 3$ grid containing the positive integers $1, 2, \\dots, 9$ arranged randomly. On a player's turn, they select one of the four $2 \\times 2$ squares of cells $S$ in the grid, containing at least two distinct numbers, and each number in $S$ is replaced by the smallest numb... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bafa318fd51da656e516ed4ca1d5ea162bbf2f86aea399b3396273d99ea62967 | Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball. If Bob has the ball, there is a 2/5 chance that he will toss it to Alice, and if he doesn't toss it to Alice, he keeps it. Alice starts with th... | [
"\\frac{9}{20}"
] | [
"Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball. If Bob has the ball, there is a 2/5 chance that he will toss it to Alice, and if he doesn't toss it to Alice, he keeps it. Alice starts wi... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f8f128e58413c334ff4f4611a1c35b19e8e9e36ecc188d0ea263ba75dae2fa5 | Alice and Bob play a game with a row of 50 coins of various values. They alternate turns, and on each turn, a player picks either the first or last coin and takes it. If Alice plays first, what is the minimum total value of coins Alice can guarantee to collect, assuming both players play optimally? | [
"625"
] | [
"Alice and Bob play a game with a row of 50 coins of various values. They alternate turns, and on each turn, a player picks either the first or last coin and takes it. If Alice plays first, what is the minimum total value of coins Alice can guarantee to collect, assuming both players play optimally?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-63223e729b1e5473a5b1c833f4f6d1a9ad9cfeb02f4c241355118a529d54327d | Alice and Bob play a game with a row of 51 coins of various values. They alternate turns, and on each turn, a player picks either the first or last coin and takes it. Find an example where Bob can end up with more money than Alice if he plays optimally. | [
"\\(x_1=x_3=...=x_{51}=1\\) and \\(x_2=x_4=...=x_{50}=10^{10^{100}}\\)"
] | [
"Alice and Bob play a game with a row of 51 coins of various values. They alternate turns, and on each turn, a player picks either the first or last coin and takes it. Find an example where Bob can end up with more money than Alice if he plays optimally."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f16b9c30b8650e60a55a333473bdb39abbd01fa3a7ad073f472deeaf92b34991 | Alice and Bob play a game with a string of 2015 pearls. In each move, one player cuts the string between two pearls, and the other player chooses one of the resulting parts of the string while the other part is discarded. Alice makes the first move, and thereafter, the players take turns. A player loses if they obtain ... | [
"Bob has the winning strategy."
] | [
"Alice and Bob play a game with a string of 2015 pearls. In each move, one player cuts the string between two pearls, and the other player chooses one of the resulting parts of the string while the other part is discarded. Alice makes the first move, and thereafter, the players take turns. A player loses if they ob... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-d12bd6fa6e29972d8aaec0978cd038fc5b1ce782bff5f0c98970c09489c2159d | Alice and Bob play a game with a string of 2017 pearls. In each move, one player cuts the string between two pearls, and the other player chooses one of the resulting parts of the string while the other part is discarded. Alice makes the first move. A player loses if they obtain a string with a single pearl such that n... | [
"Bob has a winning strategy."
] | [
"Alice and Bob play a game with a string of 2017 pearls. In each move, one player cuts the string between two pearls, and the other player chooses one of the resulting parts of the string while the other part is discarded. Alice makes the first move. A player loses if they obtain a string with a single pearl such t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 18
} | 39 | 0 | 0 |
omr-ws-e3c90f90043a563b467e7b49f18d72aa69ef449955b1e35d48cf17b6f36ae226 | Alice and Bob play a game with an unfair coin that flips heads with probability $\tfrac{2}{3}$. They keep track of $h$, the number of heads flipped, and $t$, the number of tails flipped. After every flip, Alice gains a point if $h-t=1$, and Bob gains a point if $t-h=2$. Alice wins the game if she has 2 points, while Bo... | [
"75"
] | [
"Alice and Bob play a game with an unfair coin that flips heads with probability $\\tfrac{2}{3}$. They keep track of $h$, the number of heads flipped, and $t$, the number of tails flipped. After every flip, Alice gains a point if $h-t=1$, and Bob gains a point if $t-h=2$. Alice wins the game if she has 2 points, wh... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-d540ea133a78a8003c715295f9dc760da9094fcee856ee3cf2ff86dcb50be073 | Alice and Bob play a game with four cards, each containing one of the expressions $x + y$, $x - y$, $x^2 + xy + y^2$, and $x^2 - xy + y^2$. One card is revealed, and Alice chooses two of the remaining cards, giving the other two to Bob. Alice then assigns a real value to one of the variables $x$ or $y$, and Bob assigns... | [
"Alice has a winning strategy."
] | [
"Alice and Bob play a game with four cards, each containing one of the expressions $x + y$, $x - y$, $x^2 + xy + y^2$, and $x^2 - xy + y^2$. One card is revealed, and Alice chooses two of the remaining cards, giving the other two to Bob. Alice then assigns a real value to one of the variables $x$ or $y$, and Bob as... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 3
} | 15 | 0 | 0 |
omr-ws-2c6b6745f46afe83a9940d5e881014cf5599b9ab4bbf8c5c5b27cb562b48caa4 | Alice and Bob play a game with piles of rocks. In each turn, a player can either take one rock from some pile and remove it, or choose two piles with $x$ and $y$ rocks respectively and replace them with one pile containing $xy$ rocks. The player who cannot make a move loses. Determine who has a winning strategy, given ... | [
"Alice"
] | [
"Alice and Bob play a game with piles of rocks. In each turn, a player can either take one rock from some pile and remove it, or choose two piles with $x$ and $y$ rocks respectively and replace them with one pile containing $xy$ rocks. The player who cannot make a move loses. Determine who has a winning strategy, g... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-c688dd4eb9240bb04e8869c55b9f3c4183a8f4ec7544f01025d231639066db6c | Alice and Bob play a game with three piles of stones containing $a$, $b$, and $c$ stones, respectively. On each turn, a player must select two piles with different numbers of stones and move at least one stone from the pile with more stones (pile A) to the other pile (pile B) such that the number of stones in pile A is... | [
"(a, b, c) \\mid a = b = c \\text{ or two of } a, b, c \\text{ are equal and the third differs by } 1"
] | [
"Alice and Bob play a game with three piles of stones containing $a$, $b$, and $c$ stones, respectively. On each turn, a player must select two piles with different numbers of stones and move at least one stone from the pile with more stones (pile A) to the other pile (pile B) such that the number of stones in pile... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-da87538595cbc6bcb21c6c2b7a4b17d849b45ed8bfd7f7d60f8d9e441ad3b901 | Alice and Bob play a game. Alice secretly picks a finite set \( S \) of lattice points in the Cartesian plane. For every line \( \ell \) in the plane which is horizontal, vertical, or has slope \( +1 \) or \( -1 \), she tells Bob the number of points of \( S \) that lie on \( \ell \). Bob wins if he can determine the s... | [
"0"
] | [
"Alice and Bob play a game. Alice secretly picks a finite set \\( S \\) of lattice points in the Cartesian plane. For every line \\( \\ell \\) in the plane which is horizontal, vertical, or has slope \\( +1 \\) or \\( -1 \\), she tells Bob the number of points of \\( S \\) that lie on \\( \\ell \\). Bob wins if he ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-861476f9cb9344c73f83e3ae3945724b251e238fb1109b06ece0fd01d55e66b9 | Alice and Bob play a game. Alice writes down \( n \) positive integers on a board, and then Bob erases some of them and writes a \( + \) or a \( - \) before each of the remaining numbers, and then computes their sum. Bob wins if he gets a multiple of 2005; otherwise, Alice wins. For which values of \( n \) does Alice w... | [
"Alice wins for \\( n \\leq 10 \\); Bob wins for \\( n \\geq 11 \\)."
] | [
"Alice and Bob play a game. Alice writes down \\( n \\) positive integers on a board, and then Bob erases some of them and writes a \\( + \\) or a \\( - \\) before each of the remaining numbers, and then computes their sum. Bob wins if he gets a multiple of 2005; otherwise, Alice wins. For which values of \\( n \\)... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-20b38cce613742dceb3e9acfc6c46c54ea0170baa854d203d07289ea11f6a1b2 | Alice and Bob play a game. Let $n$ be a fixed positive integer. Alice selects a connected simple graph on $n$ vertices. After this, Alice and Bob take turns selecting a vertex with at least one adjacent edge and removing all the edges adjacent to it. Bob starts. For which values of $n$ does Bob always have a strategy t... | [
"n = 2 \\text{ or } n \\text{ is odd}"
] | [
"Alice and Bob play a game. Let $n$ be a fixed positive integer. Alice selects a connected simple graph on $n$ vertices. After this, Alice and Bob take turns selecting a vertex with at least one adjacent edge and removing all the edges adjacent to it. Bob starts. For which values of $n$ does Bob always have a strat... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-f9dda227cf967845cf4e223a82dce637807bf8578d0ce576f28110b99d8de386 | Alice and Bob play a number game starting with a positive integer $n$. Each player, starting with Alice, can change the current number $k$ to either $k-1$ or $\lceil k/2 \rceil$. The player who changes $1$ to $0$ wins. Determine all values of $n$ for which Alice has a winning strategy. | [
"\\( n \\notin \\{2\\} \\cup \\{2^{2a+1}(2b+1) + 1 \\mid a \\geq 0, b > 0\\} \\cup \\{2^{2(a+1)} + 1 \\mid a \\geq 0\\} \\)"
] | [
"Alice and Bob play a number game starting with a positive integer $n$. Each player, starting with Alice, can change the current number $k$ to either $k-1$ or $\\lceil k/2 \\rceil$. The player who changes $1$ to $0$ wins. Determine all values of $n$ for which Alice has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 4
} | 7 | 0 | 0 |
omr-ws-fde770e13cf09b238137a492c7ad8e82732a56be29b52b0a5520015c344133ee | Alice and Bob play a number game starting with a positive integer $n$. They take turns changing the number; Alice goes first. Each player in turn may change the current number $k$ to either $k-1$ or $\lfloor k/2 \rfloor$. The person who changes $1$ to $0$ wins. For which initial $n$ does Alice have a winning strategy? | [
"$A=\\{8u+4,4u+3,8u+5\\}$ where $u\\in Z_{\\ge 0}$"
] | [
"Alice and Bob play a number game starting with a positive integer $n$. They take turns changing the number; Alice goes first. Each player in turn may change the current number $k$ to either $k-1$ or $\\lfloor k/2 \\rfloor$. The person who changes $1$ to $0$ wins. For which initial $n$ does Alice have a winning str... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-2ae70c877f855068fa0ff07aa7a7b7cd3f03d0cff3ff150d497b8453b2c245f4 | Alice and Bob play a series of rounds until one of them wins three rounds. The first player to three wins is declared the winner of the series. Given that the probability of Bob winning a single round is 4/9, what is the probability that Bob wins the series? | [
"\\(\\frac{256}{6561}\\)"
] | [
"Alice and Bob play a series of rounds until one of them wins three rounds. The first player to three wins is declared the winner of the series. Given that the probability of Bob winning a single round is 4/9, what is the probability that Bob wins the series?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-674538d527dddfef48655d543d119659492528dc22aaf7cdda0999c4c757098c | Alice and Bob play a series of rounds until one of them wins three rounds. The first player to three wins is declared the winner of the series. What is the probability that Bob wins the series? | [
"\\dfrac{1}{2}"
] | [
"Alice and Bob play a series of rounds until one of them wins three rounds. The first player to three wins is declared the winner of the series. What is the probability that Bob wins the series?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0203e6c10ff9c6c2a1a8d5a3d1521f5e05a3a772a5474ffc4dfdd8106f9cda8a | Alice and Bob play alternately on a given $10 \times 10$ grid with points $A$, $B$, $C$, and $D$ marked. Each player, on their turn, draws from 1 to 5 different paths joining points $A$ and $B$, moving exclusively to the right and up on the grid. Alice starts the game. A player loses if they draw a path that passes thr... | [
"4752"
] | [
"Alice and Bob play alternately on a given $10 \\times 10$ grid with points $A$, $B$, $C$, and $D$ marked. Each player, on their turn, draws from 1 to 5 different paths joining points $A$ and $B$, moving exclusively to the right and up on the grid. Alice starts the game. A player loses if they draw a path that pass... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-e7a4f268f063ed77236c255765e03c21fa21b0668cc392049f15faa597655e20 | Alice and Bob play alternately on a given board by drawing paths from point A to point B, moving exclusively to the right and up on the grid. Each player can draw between 1 and 5 new paths per turn. Alice starts the game. A player loses if they draw a path that passes through either point C or point D. If the grid is a... | [
"2"
] | [
"Alice and Bob play alternately on a given board by drawing paths from point A to point B, moving exclusively to the right and up on the grid. Each player can draw between 1 and 5 new paths per turn. Alice starts the game. A player loses if they draw a path that passes through either point C or point D. If the grid... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-491103955277bf9d684409476805b6b3b858fd25477c263113743c45044b432d | Alice and Bob play on an infinite board formed by equilateral triangles. In each turn, Alice first places a white token on an unoccupied cell, and then Bob places a black token on an unoccupied cell. Alice's goal is to eventually have $k$ white tokens on a line. Determine the maximum value of $k$ for which Alice can ac... | [
"4"
] | [
"Alice and Bob play on an infinite board formed by equilateral triangles. In each turn, Alice first places a white token on an unoccupied cell, and then Bob places a black token on an unoccupied cell. Alice's goal is to eventually have $k$ white tokens on a line. Determine the maximum value of $k$ for which Alice c... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20f8e8cd78f7fc7da57806c6e090b568e1231219aaee019100569720393167fb | Alice and Bob play the following game. It begins with a set of $1000$ $1\times 2$ rectangles. A move consists of choosing two rectangles (a rectangle may consist of one or several $1\times 2$ rectangles combined together) that share a common side length and combining those two rectangles into one rectangle along those ... | [
"Bob has a winning strategy by always mirroring Alice's moves."
] | [
"Alice and Bob play the following game. It begins with a set of $1000$ $1\\times 2$ rectangles. A move consists of choosing two rectangles (a rectangle may consist of one or several $1\\times 2$ rectangles combined together) that share a common side length and combining those two rectangles into one rectangle along... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-01422a068bc11f04b1aa58cbd11d9aa171f35f1bc17d7e9d54f9a3bd81952de1 | Alice and Bob split 5 beans into piles. They take turns removing a positive number of beans from a pile of their choice. The player to take the last bean loses. Alice plays first. How many ways are there to split the piles such that Alice has a winning strategy? | [
"6"
] | [
"Alice and Bob split 5 beans into piles. They take turns removing a positive number of beans from a pile of their choice. The player to take the last bean loses. Alice plays first. How many ways are there to split the piles such that Alice has a winning strategy?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-6bd3648a7b3fc15f554aef1a701076ce35b1b923ee40a0f8426cd6346981247e | Alice and Bob take \(a\) and \(b\) candies respectively, where \(0 \leq a, b \leq 3\), from a pile of 6 identical candies. They draw the candies one at a time, but one person may draw multiple candies in a row. In how many ways (considering order of drawing and values of \(a\) and \(b\)) can this happen? | [
"20"
] | [
"Alice and Bob take \\(a\\) and \\(b\\) candies respectively, where \\(0 \\leq a, b \\leq 3\\), from a pile of 6 identical candies. They draw the candies one at a time, but one person may draw multiple candies in a row. In how many ways (considering order of drawing and values of \\(a\\) and \\(b\\)) can this happe... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-0e633276db84cba537943d3d54d056b53710c7fb0450d0f6fe9df1cd71ac9cb7 | Alice and Bob take turns alternatively on a $2020 \times 2020$ board with Alice starting the game. In every move, each person colors a cell that has not been colored yet and is rewarded with as many points as the number of colored cells in the same row and column. When the board is completely colored, the points determ... | [
"2040200"
] | [
"Alice and Bob take turns alternatively on a $2020 \\times 2020$ board with Alice starting the game. In every move, each person colors a cell that has not been colored yet and is rewarded with as many points as the number of colored cells in the same row and column. When the board is completely colored, the points ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 4,
"tir": 2
} | 25 | 0 | 0 |
omr-ws-45cb0edf71b008d0c93a4670650273f2eabcd31afa6713e4cff32bb30d9423fa | Alice and Bob take turns performing operations on a positive integer $n$ written on a board. At each turn, a player can choose a nonzero digit from the number and replace the number with the difference between the number and the chosen digit. The player who first writes $0$ on the board wins. If Alice starts first, who... | [
"Alice has a winning strategy if the number of nonzero digits in \\( n \\) is odd; Bob has a winning strategy if the number of nonzero digits in \\( n \\) is even."
] | [
"Alice and Bob take turns performing operations on a positive integer $n$ written on a board. At each turn, a player can choose a nonzero digit from the number and replace the number with the difference between the number and the chosen digit. The player who first writes $0$ on the board wins. If Alice starts first... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8fe71b4b8e5c5bb6844266b19716962dfd7670a6be6f18f364577aba82c9328 | Alice and Bob take turns removing balls from a bag containing $10$ black balls and $10$ white balls, with Alice going first. Alice always removes a black ball if there is one, while Bob removes one of the remaining balls uniformly at random. Once all balls have been removed, the expected number of black balls which Bob... | [
"4519"
] | [
"Alice and Bob take turns removing balls from a bag containing $10$ black balls and $10$ white balls, with Alice going first. Alice always removes a black ball if there is one, while Bob removes one of the remaining balls uniformly at random. Once all balls have been removed, the expected number of black balls whic... | [
"aops_c5_contests_amp_programs"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-460caffdee57b5eb43ebe745469df70fb0db27b9d498c1ebeb65caeb0ee0d9f4 | Alice and Bob take turns throwing a die, starting with Alice. Alice stops when she rolls a six and then passes the die to Bob, who does the same. They continue this process until a total of \( n \) throws have been made. Calculate the expected number of times Alice throws the die. | [
"\\dfrac{n}{2} + \\dfrac{3}{2}\\left(1 - \\left(\\dfrac{2}{3}\\right)^n\\right)"
] | [
"Alice and Bob take turns throwing a die, starting with Alice. Alice stops when she rolls a six and then passes the die to Bob, who does the same. They continue this process until a total of \\( n \\) throws have been made. Calculate the expected number of times Alice throws the die."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-590452a395c7743c5cd4ed60aaee662dd74594b717942e9c21c73e51ff88dce4 | Alice and Carl play a game using a square sheet of paper. On each turn, the player makes a straight cut through the sheet, creating two new sheets. The sheet with the smaller area is discarded, and the larger sheet is given to the other player. The first player to receive a sheet of area less than $1$ square centimeter... | [
"2801"
] | [
"Alice and Carl play a game using a square sheet of paper. On each turn, the player makes a straight cut through the sheet, creating two new sheets. The sheet with the smaller area is discarded, and the larger sheet is given to the other player. The first player to receive a sheet of area less than $1$ square centi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-a502a26989e1c0f8e9fb89f4dd8e0514d99abbd1f85edddab75435d7058a161d | Alice and Mary found a scale and a box with weights. When they sorted the weights by mass, they discovered there are 5 different groups of weights. They noticed that for any two weights placed on the left side of the scale, they can find two other weights to place on the right side to balance the scale. Determine the m... | [
"13"
] | [
"Alice and Mary found a scale and a box with weights. When they sorted the weights by mass, they discovered there are 5 different groups of weights. They noticed that for any two weights placed on the left side of the scale, they can find two other weights to place on the right side to balance the scale. Determine ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-1181953e4bb56d90660cc1005237339afaf04f4a8f56c3b37768d6d7b5f34d8b | Alice attends a college in which classes meet only once per week. There are 30 non-overlapping classes to choose from, and 6 classes per day for Monday through Friday. Alice decides to register at random for 7 classes. What is the probability she will have class every day of the week (Monday through Friday)? | [
"\\(\\frac{114}{377}\\)",
"\\dfrac{114}{377}"
] | [
"Alice attends a college in which classes meet only once per week. There are 30 non-overlapping classes to choose from, and 6 classes per day for Monday through Friday. Alice decides to register at random for 7 classes. What is the probability she will have class every day of the week (Monday through Friday)?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 28
} | 60 | 0 | 0 |
omr-ws-abda2c0832f23ff9cc41fdedd1564baf81cef8deaf0591a0c5c580fec257cfa4 | Alice begins at the point (0,0) and drives at n units/sec on the graph of $y=4\sin(x)\cos(x)$. Bob begins at the point $(\pi,1)$ and drives at n units/sec along the graph of $y=-\cos(x)$. When Bob is at the point $(x,y)$, he will be the same distance from the origin as Alice for the first time. What is $x+y$? | [
"\\dfrac{\\pi}{2}"
] | [
"Alice begins at the point (0,0) and drives at n units/sec on the graph of $y=4\\sin(x)\\cos(x)$. Bob begins at the point $(\\pi,1)$ and drives at n units/sec along the graph of $y=-\\cos(x)$. When Bob is at the point $(x,y)$, he will be the same distance from the origin as Alice for the first time. What is $x+y$?"... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-ae2be0020a5e48f1dee545ac2f50fc153cb382cbc21a57cb89842ffbff3c4c6e | Alice buys three burgers and two sodas for $\$3.20$, and Bill buys two burgers and a soda for $\$2.00$. How many cents does a burger cost? | [
"80"
] | [
"Alice buys three burgers and two sodas for $\\$3.20$, and Bill buys two burgers and a soda for $\\$2.00$. How many cents does a burger cost?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-17c1042a31d010736ce383dc4e4b0f16bff18f199f9ba0eed274993e9460223b | Alice chooses $N$ distinct numbers between 0 and 1000 (inclusive). Bob lists down the absolute differences of all pairs $(a, b)$ such that $a \neq b$. Bob wishes to find 10 distinct numbers on his list that do not exceed 100. What is the least value of $N$ such that Bob can always accomplish this? | [
"101"
] | [
"Alice chooses $N$ distinct numbers between 0 and 1000 (inclusive). Bob lists down the absolute differences of all pairs $(a, b)$ such that $a \\neq b$. Bob wishes to find 10 distinct numbers on his list that do not exceed 100. What is the least value of $N$ such that Bob can always accomplish this?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-63e770177c95cc7eb01b0b147406688d433389a52668944a0fc70acb763a46f7 | Alice chooses a prime number \( p > 2 \) and then Bob chooses a positive integer \( n_0 \). Alice, in the first move, chooses an integer \( n_1 > n_0 \) and calculates the expression \( s_1 = n_0^{n_1} + n_1^{n_0} \); then Bob, in the second move, chooses an integer \( n_2 > n_1 \) and calculates the expression \( s_2 ... | [
"Alice"
] | [
"Alice chooses a prime number \\( p > 2 \\) and then Bob chooses a positive integer \\( n_0 \\). Alice, in the first move, chooses an integer \\( n_1 > n_0 \\) and calculates the expression \\( s_1 = n_0^{n_1} + n_1^{n_0} \\); then Bob, in the second move, chooses an integer \\( n_2 > n_1 \\) and calculates the exp... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-a0a095bb9eff3a61fa9433a3573c318cb7608cc688b2c5931ea49bd60dab182f | Alice chooses a random integer between $1$ and $n$ inclusive. Every second, she adds $1$ to her current number, $k$, with probability $\frac{n-k}{n+1}$, subtracts $1$ with probability $\frac{k}{n+1}$, and leaves it unchanged with probability $\frac{1}{n+1}$. She repeats this process $X$ times, where $X$ is a large posi... | [
"\\dfrac{\\dbinom{n}{k}}{2^n - 1}"
] | [
"Alice chooses a random integer between $1$ and $n$ inclusive. Every second, she adds $1$ to her current number, $k$, with probability $\\frac{n-k}{n+1}$, subtracts $1$ with probability $\\frac{k}{n+1}$, and leaves it unchanged with probability $\\frac{1}{n+1}$. She repeats this process $X$ times, where $X$ is a la... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-654f411aca4c40a0d8dd2097d32dfe69f0c2454287ccba8e420bc292f32b6761 | Alice chooses a secret polynomial \( p(x) \) with nonnegative integer coefficients. Bob wants to determine \( p(x) \) exactly by querying Alice for the value of \( p(x) \) for some integer \( x \) of his choice. What is the minimal number of queries Bob needs to determine \( p(x) \) exactly? | [
"2"
] | [
"Alice chooses a secret polynomial \\( p(x) \\) with nonnegative integer coefficients. Bob wants to determine \\( p(x) \\) exactly by querying Alice for the value of \\( p(x) \\) for some integer \\( x \\) of his choice. What is the minimal number of queries Bob needs to determine \\( p(x) \\) exactly?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed92fdd83cee90842d831c0d1cb2ffb37ffb6d61a429d4c6601c2e3fa5db0d70 | Alice chooses two natural numbers \(a\) and \(b\) between 1 and 9 (inclusive) such that \(a \leq b\). Alice tells Sunny the sum \(a + b\) and tells Pob the product \(a \times b\). The following conversation occurs: - Pob: I can't determine what \(a\) and \(b\) are. - Sunny: I knew that you didn't know and neither do I.... | [
"2 and 3"
] | [
"Alice chooses two natural numbers \\(a\\) and \\(b\\) between 1 and 9 (inclusive) such that \\(a \\leq b\\). Alice tells Sunny the sum \\(a + b\\) and tells Pob the product \\(a \\times b\\). The following conversation occurs:\n- Pob: I can't determine what \\(a\\) and \\(b\\) are.\n- Sunny: I knew that you didn't... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 25
} | 43 | 0 | 0 |
omr-ws-44947c6cb86d52b8fa4993033c963735ac01423729b11c0d27817ad21bcb6c74 | Alice chose five positive integers and found that their product was even. What is the maximum number of odd integers she could have chosen? | [
"4"
] | [
"Alice chose five positive integers and found that their product was even. What is the maximum number of odd integers she could have chosen?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d030f07bef30101a7e4560a913963238f226781a3d94d95c5d87d1158ba530aa | Alice collected 15 chocolates, 10 cookies, and 20 hard candies. Every day she eats 6 treats at random. What is the probability that the first day she picks 3 hard candies among her 6? | [
"\\dfrac{43700}{135751}"
] | [
"Alice collected 15 chocolates, 10 cookies, and 20 hard candies. Every day she eats 6 treats at random. What is the probability that the first day she picks 3 hard candies among her 6?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a4ed87d50eae9ce259c7639cbe1c4d53725676a674fb7597e4d87c6b1e190cb2 | Alice drew a regular $2021$-gon in the plane. Bob labeled each vertex with a real number such that the labels of consecutive vertices differ by at most $1$. Alice then drew a diagonal between every pair of non-consecutive vertices whose labels differ by at most $1$. Let $d$ be the number of diagonals Alice drew. Find t... | [
"2018"
] | [
"Alice drew a regular $2021$-gon in the plane. Bob labeled each vertex with a real number such that the labels of consecutive vertices differ by at most $1$. Alice then drew a diagonal between every pair of non-consecutive vertices whose labels differ by at most $1$. Let $d$ be the number of diagonals Alice drew. F... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-df894c164714a2380912a930fe63f69b777f5b7557867c8a5e11c82d9ea7fa97 | Alice eats 2 treats at random every morning and 4 treats at random every evening. What is the probability that on the first day exactly 3 of the 4 treats she picks in the evening are hard candies given that she ate 2 chocolates that morning? | [
"\\(\\frac{26220}{123410}\\)"
] | [
"Alice eats 2 treats at random every morning and 4 treats at random every evening. What is the probability that on the first day exactly 3 of the 4 treats she picks in the evening are hard candies given that she ate 2 chocolates that morning?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 20
} | 42 | 0 | 0 |
omr-ws-5756671df6d772ca53c56fbd5dfbcb5390ee78298295a2cd4fd05f8e86859c43 | Alice flips a fair coin until she flips her third "tails", at which point she stops. Given that she flips the coin at least five times and that the fifth flip was "heads," find the expected number of times that Alice flips the coin. | [
"\\dfrac{89}{11}"
] | [
"Alice flips a fair coin until she flips her third \"tails\", at which point she stops. Given that she flips the coin at least five times and that the fifth flip was \"heads,\" find the expected number of times that Alice flips the coin."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-7349124fc0e169bd88e9aefcc7bea588647383bf2e6116977c7ad51a87a79f2a | Alice gives Kuro 16 natural numbers \( x_1, x_2, \dots, x_{16} \). Every round, Kuro can change the numbers on the board into \( p(x_1 + x_2), p(x_2 + x_3), \dots, p(x_{16} + x_1) \), where \( p(m) \) is the smallest prime factor of \( m \). Kuro's goal is to make all the numbers on the board become 2. Let \( f(x_1, x_... | [
"16"
] | [
"Alice gives Kuro 16 natural numbers \\( x_1, x_2, \\dots, x_{16} \\). Every round, Kuro can change the numbers on the board into \\( p(x_1 + x_2), p(x_2 + x_3), \\dots, p(x_{16} + x_1) \\), where \\( p(m) \\) is the smallest prime factor of \\( m \\). Kuro's goal is to make all the numbers on the board become 2. L... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-dabe00c8925da59cdccfad90ffef1c27aea9aced26b5507c1eebf89eac81c322 | Alice has 20 dollars and Bob has 10 dollars. They play a game where they flip a coin: if it is heads, Alice gives Bob 1 dollar, and if it is tails, Bob gives Alice 1 dollar. The game ends when one person has all the money. What is the probability that Alice will have all the money when the game ends? | [
"\\(\\frac{2}{3}\\)"
] | [
"Alice has 20 dollars and Bob has 10 dollars. They play a game where they flip a coin: if it is heads, Alice gives Bob 1 dollar, and if it is tails, Bob gives Alice 1 dollar. The game ends when one person has all the money. What is the probability that Alice will have all the money when the game ends?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-5db0c01ec434515af82355201dddefeeb98fafe75e05706b0c66f2438968a873 | Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples? | [
"190"
] | [
"Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fd385b91991b486cdbae3315a5ab575ad7a9ec23a407473c0e0526384d26ed16 | Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each person has at least 2 apples? | [
"190"
] | [
"Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each person has at least 2 apples?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f2c2c627c59647bfbfba3b9e887c578f5cfa1433dd695fd5a58e130f18fdc8de | Alice has 9 distinct pairs of socks in her drawer. She picks 5 socks at random without replacement. Given that the expected value of the number of pairs of socks she draws is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$. | [
"27"
] | [
"Alice has 9 distinct pairs of socks in her drawer. She picks 5 socks at random without replacement. Given that the expected value of the number of pairs of socks she draws is $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-36249ac1bcc2c9a77630818b05703ef015c578784ca3af73245a8e73623ea66c | Alice has \( n > 1 \) one-variable quadratic polynomials written on paper, which she keeps secret from Bob. On each move, Bob announces a real number, and Alice tells him the value of one of her polynomials at this number. Let \( f(n) \) be the minimum number of questions Bob needs to ask to guarantee he can determine ... | [
"\\dfrac{7}{243}"
] | [
"Alice has \\( n > 1 \\) one-variable quadratic polynomials written on paper, which she keeps secret from Bob. On each move, Bob announces a real number, and Alice tells him the value of one of her polynomials at this number. Let \\( f(n) \\) be the minimum number of questions Bob needs to ask to guarantee he can d... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-1cf5537ae64ba1daeacd7d893ec91ac9abe16aa5c14b0c60fe7a426da92d9bdd | Alice has \( n \) (possibly empty) bags of coins labelled \( 1 \) to \( n \). At the \( k^{\text{th}} \) move, she lists all the bags which have (strictly) fewer coins than the \( k^{\text{th}} \) bag (indices reduced modulo \( n \)), but differ by (strictly) less than \( n \) coins from it. If there is at least one li... | [
"\\( n^2 - n - 1 \\)"
] | [
"Alice has \\( n \\) (possibly empty) bags of coins labelled \\( 1 \\) to \\( n \\). At the \\( k^{\\text{th}} \\) move, she lists all the bags which have (strictly) fewer coins than the \\( k^{\\text{th}} \\) bag (indices reduced modulo \\( n \\)), but differ by (strictly) less than \\( n \\) coins from it. If the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 5
} | 26 | 0 | 0 |
omr-ws-659efddc2eaa49240719be18f932d86afe52c00ed6cbbd3c42f9993c7416ca21 | Alice has a $\triangle ABC$ with $AB$, $AC$, $BC$ having side lengths of $13$, $14$, and $15$, respectively. Let the orthocenter of $\triangle ABC$ be $H$. Let $H'$ be the reflection of $H$ about the midpoint of $AC$. Then the distance $BH'$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime posi... | [
"69"
] | [
"Alice has a $\\triangle ABC$ with $AB$, $AC$, $BC$ having side lengths of $13$, $14$, and $15$, respectively. Let the orthocenter of $\\triangle ABC$ be $H$. Let $H'$ be the reflection of $H$ about the midpoint of $AC$. Then the distance $BH'$ can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively pr... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f0896d57b987c6fd383ecfdf26fc7d56718ac2bbfa07f5f2dfff4177980d498 | Alice has a 50% chance of babysitting, a 75% chance of opening her lemonade stand, a 25% chance of spending $10 at a store, and a 10% chance of buying a scratch ticket each week. She earns $76 per week babysitting, $10 in profit per week from her lemonade stand, and the scratch ticket costs $20 with a 1 in a million ch... | [
"48"
] | [
"Alice has a 50% chance of babysitting, a 75% chance of opening her lemonade stand, a 25% chance of spending $10 at a store, and a 10% chance of buying a scratch ticket each week. She earns $76 per week babysitting, $10 in profit per week from her lemonade stand, and the scratch ticket costs $20 with a 1 in a milli... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-03bc764c79a902a3c7589680a160bc755183dadbc2d6919eabceab4d4bbf50ae | Alice has a choice between rolling two standard six-sided dice and taking the maximum of the two, or rolling two standard twelve-sided dice and taking the minimum of the two. Which choice should she choose to have the greatest expected value? | [
"Alice should choose to roll two six-sided dice and take the maximum of the two."
] | [
"Alice has a choice between rolling two standard six-sided dice and taking the maximum of the two, or rolling two standard twelve-sided dice and taking the minimum of the two. Which choice should she choose to have the greatest expected value?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dba1db6ffb82312a0c8d291e0bc4ece735d9dce306e93a279856b35d3286fb30 | Alice has a circular key ring with $n$ keys, where $n \ge 3$. She cannot distinguish the keys by their orientation or position, only by their color. What is the minimum number of colors needed to ensure that she can always identify the keys uniquely, even if the key ring is rotated or flipped? | [
"\\[\n\\begin{cases} \n2 & \\text{if } n \\ge 6, \\\\\n3 & \\text{if } n = 3, 4, 5.\n\\end{cases}\n\\]"
] | [
"Alice has a circular key ring with $n$ keys, where $n \\ge 3$. She cannot distinguish the keys by their orientation or position, only by their color. What is the minimum number of colors needed to ensure that she can always identify the keys uniquely, even if the key ring is rotated or flipped?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-6e7317f15fec49e3a386415374df7c1a90c82a4c5db371ae5a0c8c2f8ed5777b | Alice has a collection of $2m-1$ different perfect matchings of $K_{m,m}$. Suppose she picks one edge from $m$ different matchings (and no edge from the other $m-1$ matchings) so that the picked edges form a perfect matching. Let $S$ be the set of all possible ways she can choose these $m$ matchings. Calculate the size... | [
"126"
] | [
"Alice has a collection of $2m-1$ different perfect matchings of $K_{m,m}$. Suppose she picks one edge from $m$ different matchings (and no edge from the other $m-1$ matchings) so that the picked edges form a perfect matching. Let $S$ be the set of all possible ways she can choose these $m$ matchings. Calculate the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-eacefbf1622218831ef5a69d86d4fc78391ef16292e6c999efc136dd27a5500c | Alice has a deck of 36 cards, 4 suits of 9 cards each. She picks any 18 cards and gives the rest to Bob. Each turn, Alice picks any of her cards and lays it face-up onto the table, then Bob similarly picks any of his cards and lays it face-up onto the table. If this pair of cards has the same suit or the same value, Bo... | [
"15"
] | [
"Alice has a deck of 36 cards, 4 suits of 9 cards each. She picks any 18 cards and gives the rest to Bob. Each turn, Alice picks any of her cards and lays it face-up onto the table, then Bob similarly picks any of his cards and lays it face-up onto the table. If this pair of cards has the same suit or the same valu... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-1980fd9b27dcb3654b35feda9335079f4aad07049a572d837bdd6d5be308e298 | Alice has a large supply of cards, each bearing one of the numbers $1, 2, \dots, 5$. She constructs a finite deck of cards and shows it to Bob. Alice then randomly shuffles the deck. Bob guesses the value of the top card in the deck. If he guesses correctly, he wins $x$ dollars, where $x$ is his guess. How should Alice... | [
"$\\frac{60}{137}$"
] | [
"Alice has a large supply of cards, each bearing one of the numbers $1, 2, \\dots, 5$. She constructs a finite deck of cards and shows it to Bob. Alice then randomly shuffles the deck. Bob guesses the value of the top card in the deck. If he guesses correctly, he wins $x$ dollars, where $x$ is his guess. How should... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 11
} | 25 | 0 | 0 |
omr-ws-3e0318c541d0d384fbb4a2a39c61215f97dd0ecf1e0314f2593fd3134dae9e0c | Alice has a map of Wonderland, a country consisting of $n \geq 2$ towns, where each pair of towns is connected by a one-way road. Alice can ask the King of Hearts the direction of the road between any pair of towns. Let $Q(n)$ be the minimum number of questions Alice needs to ask to guarantee she can determine if there... | [
"9"
] | [
"Alice has a map of Wonderland, a country consisting of $n \\geq 2$ towns, where each pair of towns is connected by a one-way road. Alice can ask the King of Hearts the direction of the road between any pair of towns. Let $Q(n)$ be the minimum number of questions Alice needs to ask to guarantee she can determine if... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aae3e613919ca93606c5a4e32cf7e9677f10b58a78a04132e04e327dbc5de437 | Alice has a rectangular room with $6$ outlets lined up on one wall and $6$ lamps lined up on the opposite wall. She has $6$ distinct power cords (red, blue, green, purple, black, yellow). If the red and green power cords cannot cross, how many ways can she plug in all six lamps? | [
"360"
] | [
"Alice has a rectangular room with $6$ outlets lined up on one wall and $6$ lamps lined up on the opposite wall. She has $6$ distinct power cords (red, blue, green, purple, black, yellow). If the red and green power cords cannot cross, how many ways can she plug in all six lamps?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-68c84a7b463fe41bcc9e8f0ba444e1a6da8420ffa8e92efbdaa2e5bb36ec37ac | Alice has a whiteboard and a blackboard. The whiteboard has two positive integers on it, and the blackboard is initially blank. Alice repeats the following process: - Let the numbers on the whiteboard be \(a\) and \(b\), with \(a \le b\). - Write \(a^2\) on the blackboard. - Erase \(b\) from the whiteboard and replace ... | [
"(31, 65)"
] | [
"Alice has a whiteboard and a blackboard. The whiteboard has two positive integers on it, and the blackboard is initially blank. Alice repeats the following process:\n- Let the numbers on the whiteboard be \\(a\\) and \\(b\\), with \\(a \\le b\\).\n- Write \\(a^2\\) on the blackboard.\n- Erase \\(b\\) from the whit... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-8cf5324203d3c31b27d80321dc8ba7587f5136cbf58de5395e406f4d2257f107 | Alice has an integer \( N > 1 \) on the blackboard. Each minute, she deletes the current number \( x \) on the blackboard and writes \( 2x + 1 \) if \( x \) is not the cube of an integer, or the cube root of \( x \) otherwise. Let \( T(N) \) be the minimum number of minutes required for the number on the blackboard to ... | [
"331"
] | [
"Alice has an integer \\( N > 1 \\) on the blackboard. Each minute, she deletes the current number \\( x \\) on the blackboard and writes \\( 2x + 1 \\) if \\( x \\) is not the cube of an integer, or the cube root of \\( x \\) otherwise. Let \\( T(N) \\) be the minimum number of minutes required for the number on t... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b008c8f0a856443312b6bc043601b0757e36c25c5de6f0296e1d8588d1916a99 | Alice has an object that starts at the origin in an infinite coordinate plane. Every turn, she rolls a fair six-sided die. If she rolls a 1 or 2, she moves the object up one unit. If she rolls a 3, 4, or 5, she moves the object right one unit. If she rolls a 6, she removes the object from the plane. Let \( C(p) \) be t... | [
"6"
] | [
"Alice has an object that starts at the origin in an infinite coordinate plane. Every turn, she rolls a fair six-sided die. If she rolls a 1 or 2, she moves the object up one unit. If she rolls a 3, 4, or 5, she moves the object right one unit. If she rolls a 6, she removes the object from the plane. Let \\( C(p) \... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a93fedf8f6e9bc484880b00bfd6f980fe296d1dcc50ee828ecdef515a8826518 | Alice has an unfair dice such that the probability of rolling a $1$, $2$, $3$, $4$, $5$, $6$ forms an arithmetic sequence, in this order. Bob rolls the dice twice, getting $a$ and $b$ respectively. There is a $\frac{1}{7}$ probability that $a+b=7$. Find the probability that $a=b$. | [
"\\(\\frac{4}{21}\\)"
] | [
"Alice has an unfair dice such that the probability of rolling a $1$, $2$, $3$, $4$, $5$, $6$ forms an arithmetic sequence, in this order. Bob rolls the dice twice, getting $a$ and $b$ respectively. There is a $\\frac{1}{7}$ probability that $a+b=7$. Find the probability that $a=b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 22
} | 70 | 0 | 0 |
omr-ws-43e3313fe23fa24597cae880e9d594f1a0deceacc394b6731af50803c9b1b8b3 | Alice has chosen a number from 1 to 16. Bob can ask seven yes-no questions to determine the number, with Alice allowed to give at most one wrong answer. How can Bob devise a strategy to identify the number in all cases? | [
"7"
] | [
"Alice has chosen a number from 1 to 16. Bob can ask seven yes-no questions to determine the number, with Alice allowed to give at most one wrong answer. How can Bob devise a strategy to identify the number in all cases?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-24c86a31854029e68d5503ac667385c641479854ed58682429695934b9ffb5d3 | Alice has four boxes, 327 blue balls labeled from 1 to 327, and 2022 red balls. She distributes all the balls into the boxes, possibly leaving some boxes empty. A random label between 1 and 327 is selected, Alice finds the box containing the ball with that label, and then selects a random ball from that box. What is th... | [
"1"
] | [
"Alice has four boxes, 327 blue balls labeled from 1 to 327, and 2022 red balls. She distributes all the balls into the boxes, possibly leaving some boxes empty. A random label between 1 and 327 is selected, Alice finds the box containing the ball with that label, and then selects a random ball from that box. What ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b22d08a755866a1dc1d8b0809068fe7c016ca7c9bf6f3c654f9e01c74b78edb | Alice has six boxes labelled 1 through 6. She secretly chooses exactly two of the boxes and places a coin inside each. Bob guesses by tapping exactly two boxes each time, and Alice responds with the total number of coins in the boxes he tapped. Bob successfully finds the two coins when Alice responds with the number 2.... | [
"5"
] | [
"Alice has six boxes labelled 1 through 6. She secretly chooses exactly two of the boxes and places a coin inside each. Bob guesses by tapping exactly two boxes each time, and Alice responds with the total number of coins in the boxes he tapped. Bob successfully finds the two coins when Alice responds with the numb... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-f11ec38736639150e91ee21da43ad6009306076b3f02625e3fcfcefb515d276d | Alice has two bags of marbles: Bag A contains 8 red and 3 blue marbles, and Bag B contains 5 red and 7 blue marbles. She randomly selects a bag and then draws one marble at random. Given that the marble drawn is blue, what is the probability that it came from Bag B? | [
"\\( \\frac{77}{113} \\)"
] | [
"Alice has two bags of marbles: Bag A contains 8 red and 3 blue marbles, and Bag B contains 5 red and 7 blue marbles. She randomly selects a bag and then draws one marble at random. Given that the marble drawn is blue, what is the probability that it came from Bag B?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-8a0279c420a756f075d0f6eb33105a1be689a9408e406a7596234098a94d3047 | Alice has two bags of marbles: Bag A contains 8 red and 3 blue marbles, and Bag B contains 5 red and 7 blue marbles. She randomly selects a bag and then draws one marble at random. If the marble drawn is blue, what is the probability that it came from Bag B? | [
"\\(\\frac{77}{113}\\)"
] | [
"Alice has two bags of marbles: Bag A contains 8 red and 3 blue marbles, and Bag B contains 5 red and 7 blue marbles. She randomly selects a bag and then draws one marble at random. If the marble drawn is blue, what is the probability that it came from Bag B?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5dffffecc57a8a38baa50323e99fd0b789176c512843eeb2743ab788267b58e5 | Alice has two urns, each containing four balls with natural numbers written on them. She draws one ball from each urn, notes the sum of the numbers, and repeats this process many times. Bill observes that the frequency of each sum is the same as if the numbers were drawn at random from the range 1 to 4. If the numbers ... | [
"24"
] | [
"Alice has two urns, each containing four balls with natural numbers written on them. She draws one ball from each urn, notes the sum of the numbers, and repeats this process many times. Bill observes that the frequency of each sum is the same as if the numbers were drawn at random from the range 1 to 4. If the num... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c8bbbfcbab7964de07f73376028b153fdf59777d4d8f4f1e81efab54367e8de | Alice is at point $A$ and Charlie is at point $C$ of square $ABCD$. Each second, both Alice and Charlie move to one of the neighboring vertices with equal probability. What is the expected number of seconds until Alice and Charlie meet each other? | [
"2"
] | [
"Alice is at point $A$ and Charlie is at point $C$ of square $ABCD$. Each second, both Alice and Charlie move to one of the neighboring vertices with equal probability. What is the expected number of seconds until Alice and Charlie meet each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c8269c2b3d91c78b42ad1f3d57acf824f54ac6cab38aefff0ac8995e3c15659f | Alice is at the origin (0, 0) of the coordinate plane. She moves according to the roll of a standard cubical die. If the die rolls a 1, she moves 1 space to the right. If the die rolls a 2 or 3, she moves 1 space to the left. If the die rolls a 4, 5, or 6, she moves 1 space up. What is the probability that after four m... | [
"\\(\\frac{1}{27}\\)"
] | [
"Alice is at the origin (0, 0) of the coordinate plane. She moves according to the roll of a standard cubical die. If the die rolls a 1, she moves 1 space to the right. If the die rolls a 2 or 3, she moves 1 space to the left. If the die rolls a 4, 5, or 6, she moves 1 space up. What is the probability that after f... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 43
} | 73 | 0 | 0 |
omr-ws-172d6ab5ce013266381f00de6fbd8938cb1e3c1de63193017d8db9205954bac1 | Alice is currently at level $7$ and Bob is at level $8$. The probability of levelling up once is $\frac{1}{4}$, the probability of remaining at the same level is $\frac{1}{2}$, and the probability of falling back to the previous level is $\frac{1}{4}$. What is the difference between the expected number of attempts for ... | [
"4"
] | [
"Alice is currently at level $7$ and Bob is at level $8$. The probability of levelling up once is $\\frac{1}{4}$, the probability of remaining at the same level is $\\frac{1}{2}$, and the probability of falling back to the previous level is $\\frac{1}{4}$. What is the difference between the expected number of attem... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bb6122d9d02b300b24feb3f60ea6e554c4b96c9a0d88d05040d08da807ba1360 | Alice is filming two trick-shots: trick-shot $A$ takes 10 seconds to film and has a $\frac{1}{10}$ success rate, and trick-shot $B$ takes 20 seconds to film and has a $\frac{1}{4}$ success rate. If she succeeds in both, she celebrates for 10 seconds. On average, how much faster is each video if Alice records the trick-... | [
"10.5"
] | [
"Alice is filming two trick-shots: trick-shot $A$ takes 10 seconds to film and has a $\\frac{1}{10}$ success rate, and trick-shot $B$ takes 20 seconds to film and has a $\\frac{1}{4}$ success rate. If she succeeds in both, she celebrates for 10 seconds. On average, how much faster is each video if Alice records the... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-bdc24f0fbcd2bc62b9c6a6f5f62752da21be02513b4685b5b8079144521b83dc | Alice is going from (0, 0) to (n, n). In each step, she either moves one unit to the right or one unit upwards. Find the number of paths for Alice that never cross the line y = x. | [
"\\dfrac{1}{n+1} \\dbinom{2n}{n}"
] | [
"Alice is going from (0, 0) to (n, n). In each step, she either moves one unit to the right or one unit upwards. Find the number of paths for Alice that never cross the line y = x."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7a95291cc881651fa8cd36335b9c52d10ac32cab38788e8950430abe23d276a2 | Alice is in a maze designed by Bob, which is a circular sequence of $n$ pairwise distinct positive real numbers $x_1, x_2, \cdots, x_n$. Alice goes through all $1 \leq i \leq n$ and writes down the angle opposite the side with length $a_i$ in the triangle with side lengths $a_{i-1}, a_i, a_{i+1}$. Alice's score is the ... | [
"n-2"
] | [
"Alice is in a maze designed by Bob, which is a circular sequence of $n$ pairwise distinct positive real numbers $x_1, x_2, \\cdots, x_n$. Alice goes through all $1 \\leq i \\leq n$ and writes down the angle opposite the side with length $a_i$ in the triangle with side lengths $a_{i-1}, a_i, a_{i+1}$. Alice's score... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-0a5228fef26b673bd034085fb4990197253f9c636c275d4bb819c9bdf4d2d72f | Alice is playing a game with $2018$ boxes, numbered $1$ to $2018$, and a number of balls. At the beginning, boxes $1$ to $2017$ each have one ball, and box $2018$ has $2018n$ balls. Every turn, Alice chooses $i$ and $j$ with $i > j$, and moves exactly $i$ balls from box $i$ to box $j$. Alice wins if all balls end up in... | [
"2016"
] | [
"Alice is playing a game with $2018$ boxes, numbered $1$ to $2018$, and a number of balls. At the beginning, boxes $1$ to $2017$ each have one ball, and box $2018$ has $2018n$ balls. Every turn, Alice chooses $i$ and $j$ with $i > j$, and moves exactly $i$ balls from box $i$ to box $j$. Alice wins if all balls end ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-abd8331265a877e0254f7d5a938f73716bf4d93235c9ce5d6b517d12d5593ea6 | Alice is standing 5 meters away from Bob. Bob is standing 20 meters away from Charlie. What is the minimum possible distance between Alice and Charlie in meters? | [
"15"
] | [
"Alice is standing 5 meters away from Bob. Bob is standing 20 meters away from Charlie. What is the minimum possible distance between Alice and Charlie in meters?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c993928bbb3b70741b82d38fff625c3824d8a69423d7491bfb88b7b9b76644e5 | Alice is standing on the circumference of a large circular room of radius $10$. There is a circular pillar in the center of the room of radius $5$ that blocks Alice’s view. The total area in the room Alice can see can be expressed in the form $\frac{m\pi}{n} + p\sqrt{q}$, where $m$ and $n$ are relatively prime positive... | [
"231"
] | [
"Alice is standing on the circumference of a large circular room of radius $10$. There is a circular pillar in the center of the room of radius $5$ that blocks Alice’s view. The total area in the room Alice can see can be expressed in the form $\\frac{m\\pi}{n} + p\\sqrt{q}$, where $m$ and $n$ are relatively prime ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-ed3e693332bf12edd6f09b28d5ace2057d5b6b82b95a7b67beabca2bc09df7c4 | Alice is throwing a coin and continues until she gets the flip side or until she has thrown it \( n \) times. Calculate the expected number of throws. | [
"\\( 2 \\left(1 - \\frac{1}{2^n}\\right) \\)"
] | [
"Alice is throwing a coin and continues until she gets the flip side or until she has thrown it \\( n \\) times. Calculate the expected number of throws."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-9ce9b63ff25df75dbfdef66050645bd6cc2d65b6eaba8fe427ae447a1a088c70 | Alice is walking at a constant speed of $40$ meters per minute west. Bob is $150$ meters north of Alice and walks at a uniform pace of $50$ meters per minute in a straight line such that he will eventually meet Alice. Suppose that Bob is walking at an angle $\theta$ south of west, and their meeting will occur in $t$ mi... | [
"3"
] | [
"Alice is walking at a constant speed of $40$ meters per minute west. Bob is $150$ meters north of Alice and walks at a uniform pace of $50$ meters per minute in a straight line such that he will eventually meet Alice. Suppose that Bob is walking at an angle $\\theta$ south of west, and their meeting will occur in ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-08478b1ed6660af8b0baa11adda116f0c9a90c0e791a08dc4afb63ac6d1a4c47 | Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. ... | [
"051"
] | [
"Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\\frac{m}{n},$ where $m$ and $n$ are relatively prime positive inte... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 18
} | 38 | 0 | 0 |
omr-ws-ae647161affd7b2004a63b8c878a8c9a20d898026c5acb44a4089771a3a04831 | Alice knows that 3 red cards and 3 black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find... | [
"051"
] | [
"Alice knows that 3 red cards and 3 black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-282e18cae1e92d5fbb69c6727ed1516a26d4f9cc055a9925e121a1430e390f90 | Alice knows that 3 red cards and 3 black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find... | [
"51"
] | [
"Alice knows that 3 red cards and 3 black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-752689b86812a35e83316e3bf0c97822b87bce3b39c84d8c5ffa793b2a57188f | Alice needs to move from (0, 0) to (5, 5), moving either one unit up or one unit to the right at each step. However, she cannot walk from (2, 2) to (2, 3) because a wall is in the way. Find the number of different paths for Alice. | [
"192"
] | [
"Alice needs to move from (0, 0) to (5, 5), moving either one unit up or one unit to the right at each step. However, she cannot walk from (2, 2) to (2, 3) because a wall is in the way. Find the number of different paths for Alice."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-0a0365b175327154bad59fc574373d5934642fe01e530bfee8520b71245e493b | Alice organized the numbers from $1$ to $100^2$ in a $100 \times 100$ table, ensuring that each column is in ascending order from top to bottom, and each row is in descending order from left to right. Determine the maximum value that the sum of the numbers in the corners of the table can achieve. | [
"20002"
] | [
"Alice organized the numbers from $1$ to $100^2$ in a $100 \\times 100$ table, ensuring that each column is in ascending order from top to bottom, and each row is in descending order from left to right. Determine the maximum value that the sum of the numbers in the corners of the table can achieve."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-4cd31a7c11ef56c7b90403611eda3fe6170b7f91a0282a40acb43ae4891d4116 | Alice picked positive integers \(a, b, c\) and found unique positive integers \(x, y, z\) such that \(a = \text{lcm}(x, y)\), \(b = \text{lcm}(x, z)\), and \(c = \text{lcm}(y, z)\). Given \(a = 120\) and \(b = 180\), calculate the value of \(c\). | [
"360"
] | [
"Alice picked positive integers \\(a, b, c\\) and found unique positive integers \\(x, y, z\\) such that \\(a = \\text{lcm}(x, y)\\), \\(b = \\text{lcm}(x, z)\\), and \\(c = \\text{lcm}(y, z)\\). Given \\(a = 120\\) and \\(b = 180\\), calculate the value of \\(c\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-33aef36616a90fbe782cb7daafd20aafcf8fc3edfe01ca955497e5303cd18129 | Alice picks a number between 0 and 1, and Bob picks a number between 0 and 1/2. If the probability that Alice's number is greater than Bob's number can be expressed as \(\frac{m}{n}\) for relatively prime positive integers \(m\) and \(n\), find \(m+n\). | [
"7"
] | [
"Alice picks a number between 0 and 1, and Bob picks a number between 0 and 1/2. If the probability that Alice's number is greater than Bob's number can be expressed as \\(\\frac{m}{n}\\) for relatively prime positive integers \\(m\\) and \\(n\\), find \\(m+n\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a7ad2db443c994ea402d5b30d3aeeea41af146aee354d04858dd89979c4222fa | Alice picks a number uniformly at random from the first 5 even positive integers, and Palice picks a number uniformly at random from the first 5 odd positive integers. If Alice picks a larger number than Palice with probability \(\frac{m}{n}\) for relatively prime positive integers \(m\) and \(n\), compute \(m+n\). | [
"8"
] | [
"Alice picks a number uniformly at random from the first 5 even positive integers, and Palice picks a number uniformly at random from the first 5 odd positive integers. If Alice picks a larger number than Palice with probability \\(\\frac{m}{n}\\) for relatively prime positive integers \\(m\\) and \\(n\\), compute ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e2adcfe749b4dd0cc061fa067302581a162556506036f8035651f2ebd19e244e | Alice picks a polynomial $p(x)$ with non-negative integer coefficients. Bob can ask her the value of $p(n)$ for any positive integer $n$ of his choice. What is the minimum number of questions Bob needs to ask Alice to determine $p(x)$ completely? | [
"2"
] | [
"Alice picks a polynomial $p(x)$ with non-negative integer coefficients. Bob can ask her the value of $p(n)$ for any positive integer $n$ of his choice. What is the minimum number of questions Bob needs to ask Alice to determine $p(x)$ completely?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7da5916d186675567acaa02f3afc44442a2ef814fb3cb60dc78182a91ec96997 | Alice picks a positive integer \( a \) such that \( a \leq 2006 \). In each turn, Bob gives Alice a positive integer \( b \) and Alice tells Bob whether or not \( a + b \) is prime. Bob wants to determine \( a \) in the minimum number of turns. If Bob uses an optimal strategy, what is the maximum number of turns Bob ne... | [
"11"
] | [
"Alice picks a positive integer \\( a \\) such that \\( a \\leq 2006 \\). In each turn, Bob gives Alice a positive integer \\( b \\) and Alice tells Bob whether or not \\( a + b \\) is prime. Bob wants to determine \\( a \\) in the minimum number of turns. If Bob uses an optimal strategy, what is the maximum number... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38260f3e89f0d06e65c6d3a049e7415bd408e7c9ab2498a1582bde299fc31ec9 | Alice picks a random number between 0 and 0.75. Charlie picks a random number between 0.25 and 1. Bob picks a random number between 0 and 1. What is the probability that Bob’s number is between Charlie and Alice’s numbers? | [
"\\(\\frac{37}{128}\\)"
] | [
"Alice picks a random number between 0 and 0.75. Charlie picks a random number between 0.25 and 1. Bob picks a random number between 0 and 1. What is the probability that Bob’s number is between Charlie and Alice’s numbers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.